Fractional cosmology: Background dynamics and first order perturbations

Various approches for constructing fractional cosmological models with additional dynamical degrees of freedom are presented. Focusing on one of these approaches, the corresponding fractional equations of motion are explicitly derived. In this fractional framework, the standard continuity equation of the corresponding relativistic cosmological model is no longer preserved, as energy is continuously exchanged between the fractional sector and the additional dynamical components. To address this issue, effective energy density and pressure are consistently defined in such a way that an effective continuity equation is satisfied. This procedure plays a crucial role not only in obtaining an appropriate dynamical system for the background evolution, but also in the systematic derivation of the perturbation equations.

The differences, advantages, and limitations of our fractional framework are examined in direct comparison with the corresponding standard cosmological models, both at the level of the field equations and their exact solutions. This comparative analysis is carried out not only at the background level, but also within first-order cosmological perturbation theory. In particular, the fractional generalization of the Mukhanov–Sasaki equation is derived, providing a consistent description of scalar perturbations in the presence of fractional effects and memory contributions. The resulting framework offers a coherent extension of relativistic cosmology in which fractional dynamics naturally influence both the background evolution and the perturbative sector.